GATE AG 2009 • Q1 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Matrices and determinants

Question Statement:

Inverse of the matrix $\begin{bmatrix} 2 & 3 \\ 2 & 1 \end{bmatrix}$ is

Options:

  • (A) $\begin{bmatrix} -0.5 & 0.75 \\ 0.5 & -0.25 \end{bmatrix}$
  • (B) $\begin{bmatrix} -0.25 & 0.5 \\ -0.5 & 0.75 \end{bmatrix}$
  • (C) $\begin{bmatrix} -0.25 & 0.75 \\ 0.5 & -0.5 \end{bmatrix}$
  • (D) $\begin{bmatrix} -0.25 & -0.5 \\ 0.75 & 0.5 \end{bmatrix}$

Official Answer Key:

C

Step-by-Step Solution & Explanation

For A = [[2,3],[2,1]], det(A) = 2(1) - 3(2) = 2 - 6 = -4. Adjugate = [[1,-3],[-2,2]]. A^-1 = (1/det)*adj = (1/-4)*[[1,-3],[-2,2]] = [[-0.25, 0.75],[0.5, -0.5]]. This matches option (C). Option (A) has the correct magnitudes but the sign pattern of a cofactor swap is wrong; (B) and (D) come from mis-placing the off-diagonal terms during adjugate formation.

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